This is an extended text of the address at the discussion on teaching
of mathematics in Palais de Découverte in Paris on 7 March 1997.) It
can be found at
http://pauli.uni-muenster.de/~munsteg/arnold.html
I think this will be of some interest to the FOM list, as it gives
insight into the thinking of a very well known and highly respected
core mathematician.
For instance, note the opening sentence. Whatever f.o.m. is, it is
difficult to say that f.o.m. is a part of physics, or a part of a
part of physics.
On the other hand, there are certain aspects of Arnold's attitudes
that I construe as f.o.m. friendly - although the text as a whole has
the appearance of being not f.o.m. friendly.
Obviously, Arnold touches on issues in philosophy of math such as in
paragraph 2. Of course, one cannot take this text as a scholarly
contribution to philosophy as their is no attempt at detailed
justifications or explanations.
On teaching mathematics
by V.I. Arnold
Mathematics is a part of physics. Physics is an experimental science,
a part of natural science. Mathematics is the part of physics where
experiments are cheap.
The Jacobi identity (which forces the heights of a triangle to cross at
one point) is an experimental fact in the same way as that the Earth
is round (that is, homeomorphic to a ball). But it can be discovered
with less expense.
In the middle of the twentieth century it was attempted to divide physics
and mathematics. The consequences turned out to be catastrophic.
Whole generations of mathematicians grew up without knowing half of their
science and, of course, in total ignorance of any other sciences.
They first began teaching their ugly scholastic pseudo-mathematics
to their students, then to schoolchildren (forgetting Hardy's warning
that ugly mathematics has no permanent place under the Sun).
Since scholastic mathematics that is cut off from physics is fit neither
for teaching nor for application in any other science, the result was
the universal hate towards mathematicians - both on the part of the poor
schoolchildren (some of whom in the meantime became ministers) and of
the users.
The ugly building, built by undereducated mathematicians who were exhausted
by their inferiority complex and who were unable to make themselves familiar
with physics, reminds one of the rigorous axiomatic theory of odd numbers.
Obviously, it is possible to create such a theory and make pupils admire
the perfection and internal consistency of the resulting structure
(in which, for example, the sum of an odd number of terms and the product
of any number of factors are defined). From this sectarian point of view,
even numbers could either be declared a heresy or, with passage of time,
be introduced into the theory supplemented with a few "ideal" objects
(in order to comply with the needs of physics and the real world).
Unfortunately, it was an ugly twisted construction of mathematics like the
one above which predominated in the teaching of mathematics for decades.
Having originated in France, this pervertedness quickly spread to teaching
of foundations of mathematics, first to university students, then to school
pupils of all lines (first in France, then in other countries, including
Russia).
To the question "what is 2 + 3" a French primary school pupil replied:
"3 + 2, since addition is commutative". He did not know what the sum was
equal to and could not even understand what he was asked about!
Another French pupil (quite rational, in my opinion) defined mathematics
as follows: "there is a square, but that still has to be proved".
Judging by my teaching experience in France, the university students'
idea of mathematics (even of those taught mathematics at the
Ecole Normale Superieure - I feel sorry most of all for these obviously
intelligent but deformed kids) is as poor as that of this pupil.
For example, these students have never seen a paraboloid and a question
on the form of the surface given by the equation xy = z^2 puts the
mathematicians studying at ENS into a stupor. Drawing a curve given by
parametric equations (like x = t^3 - 3t, y = t^4 - 2t^2) on a plane is
a totally impossible problem for students (and, probably, even for
most French professors of mathematics).
Beginning with l'Hospital's first textbook on calculus ("calculus for
understanding of curved lines") and roughly until Goursat's textbook,
the ability to solve such problems was considered to be (along with
the knowledge of the times table) a necessary part of the craft of
every mathematician.
Mentally challenged zealots of "abstract mathematics" threw all the
geometry (through which connection with physics and reality most
often takes place in mathematics) out of teaching. Calculus textbooks
by Goursat, Hermite, Picard were recently dumped by the student
library of the Universities Paris 6 and 7 (Jussieu) as obsolete and,
therefore, harmful (they were only rescued by my intervention).
ENS students who have sat through courses on differential and algebraic
geometry (read by respected mathematicians) turned out be acquainted
neither with the Riemann surface of an elliptic curve y^2 = x^3 + ax + b
nor, in fact, with the topological classification of surfaces (not even
mentioning elliptic integrals of first kind and the group property of an
elliptic curve, that is, the Euler-Abel addition theorem). They were
only taught Hodge structures and Jacobi varieties!
How could this happen in France, which gave the world Lagrange and Laplace,
Cauchy and Poincaré, Leray and Thom? It seems to me that a reasonable
explanation was given by I.G. Petrovskii, who taught me in 1966: genuine
mathematicians do not gang up, but the weak need gangs in order to survive.
They can unite on various grounds (it could be super-abstractness,
anti-Semitism or "applied and industrial" problems), but the essence is
always a solution of the social problem - survival in conditions of more
literate surroundings.
By the way, I shall remind you of a warning of L. Pasteur: there never
have been and never will be any "applied sciences", there are only
applications of sciences (quite useful ones!).
In those times I was treating Petrovskii's words with some doubt, but
now I am being more and more convinced of how right he was. A considerable
part of the super-abstract activity comes down simply to industrialising
shameless grabbing of discoveries from discoverers and then systematically
assigning them to epigons-generalizers. Similarly to the fact that America
does not carry Columbus's name, mathematical results are almost never
called by the names of their discoverers.
In order to avoid being misquoted, I have to note that my own achievements
were for some unknown reason never expropriated in this way, although it
always happened to both my teachers (Kolmogorov, Petrovskii, Pontryagin,
Rokhlin) and my pupils. Prof. M. Berry once formulated the following two
principles:
The Arnold Principle. If a notion bears a personal name, then this
name is not the name of the discoverer.
The Berry Principle. The Arnold Principle is applicable to itself.
Let's return, however, to teaching of mathematics in France.
When I was a first-year student at the Faculty of Mechanics and Mathematics
of the Moscow State University, the lectures on calculus were read by the
set-theoretic topologist L.A. Tumarkin, who conscientiously retold the old
classical calculus course of French type in the Goursat version. He told us
that integrals of rational functions along an algebraic curve can be taken
if the corresponding Riemann surface is a sphere and, generally speaking,
cannot be taken if its genus is higher, and that for the sphericity it is
enough to have a sufficiently large number of double points on the curve of
a given degree (which forces the curve to be unicursal: it is possible to
draw its real points on the projective plane with one stroke of a pen).
These facts capture the imagination so much that (even given without any
proofs) they give a better and more correct idea of modern mathematics
than whole volumes of the Bourbaki treatise. Indeed, here we find out about
the existence of a wonderful connection between things which seem to be
completely different: on the one hand, the existence of an explicit
expression for the integrals and the topology of the corresponding Riemann
surface and, on the other hand, between the number of double points and
genus of the corresponding Riemann surface, which also exhibits itself
in the real domain as the unicursality.
Jacobi noted, as mathematics' most fascinating property, that in it one and
the same function controls both the presentations of a whole number as a
sum of four squares and the real movement of a pendulum.
These discoveries of connections between heterogeneous mathematical objects
can be compared with the discovery of the connection between electricity
and magnetism in physics or with the discovery of the similarity between
the east coast of America and the west coast of Africa in geology.
The emotional significance of such discoveries for teaching is difficult
to overestimate. It is they who teach us to search and find such wonderful
phenomena of harmony of the Universe.
The de-geometrisation of mathematical education and the divorce from physics
sever these ties. For example, not only students but also modern
algebro-geometers on the whole do not know about the Jacobi fact mentioned
here: an elliptic integral of first kind expresses the time of motion
along an elliptic phase curve in the corresponding Hamiltonian system.
Rephrasing the famous words on the electron and atom, it can be said that
a hypocycloid is as inexhaustible as an ideal in a polynomial ring. But
teaching ideals to students who have never seen a hypocycloid is as
ridiculous as teaching addition of fractions to children who have never
cut (at least mentally) a cake or an apple into equal parts. No wonder
that the children will prefer to add a numerator to a numerator and a
denominator to a denominator.
>From my French friends I heard that the tendency towards super-abstract
generalizations is their traditional national trait. I do not entirely
disagree that this might be a question of a hereditary disease, but I
would like to underline the fact that I borrowed the cake-and-apple
example from Poincare.
The scheme of construction of a mathematical theory is exactly the same as
that in any other natural science. First we consider some objects and make
some observations in special cases. Then we try and find the limits of
application of our observations, look for counter-examples which would
prevent unjustified extension of our observations onto a too wide range
of events (example: the number of partitions of consecutive odd numbers
1, 3, 5, 7, 9 into an odd number of natural summands gives the sequence
1, 2, 4, 8, 16, but then comes 29).
As a result we formulate the empirical discovery that we made (for
example, the Fermat conjecture or Poincare conjecture) as clearly as
possible. After this there comes the difficult period of checking as
to how reliable are the conclusions.
At this point a special technique has been developed in mathematics.
This technique, when applied to the real world, is sometimes useful,
but can sometimes also lead to self-deception. This technique is called
modelling. When constructing a model, the following idealisation is
made: certain facts which are only known with a certain degree of
probability or with a certain degree of accuracy, are considered to be
"absolutely" correct and are accepted as "axioms". The sense of this
"absoluteness" lies precisely in the fact that we allow ourselves to
use these "facts" according to the rules of formal logic, in the
process declaring as "theorems" all that we can derive from them.
It is obvious that in any real-life activity it is impossible to wholly
rely on such deductions. The reason is at least that the parameters of
the studied phenomena are never known absolutely exactly and a small
change in parameters (for example, the initial conditions of a process)
can totally change the result. Say, for this reason a reliable
long-term weather forecast is impossible and will remain impossible,
no matter how much we develop computers and devices which record
initial conditions.
In exactly the same way a small change in axioms (of which we cannot be
completely sure) is capable, generally speaking, of leading to completely
different conclusions than those that are obtained from theorems which
have been deduced from the accepted axioms. The longer and fancier is the
chain of deductions ("proofs"), the less reliable is the final result.
Complex models are rarely useful (unless for those writing their
dissertations).
The mathematical technique of modelling consists of ignoring this
trouble and speaking about your deductive model in such a way as if it
coincided with reality. The fact that this path, which is obviously
incorrect from the point of view of natural science, often leads to
useful results in physics is called "the inconceivable effectiveness
of mathematics in natural sciences" (or "the Wigner principle").
Here we can add a remark by I.M. Gel'fand: there exists yet another
phenomenon which is comparable in its inconceivability with the
inconceivable effectiveness of mathematics in physics noted by Wigner
- this is the equally inconceivable ineffectiveness of mathematics
in biology.
"The subtle poison of mathematical education" (in F. Klein's words)
for a physicist consists precisely in that the absolutised model
separates from the reality and is no longer compared with it.
Here is a simple example: mathematics teaches us that the solution
of the Malthus equation dx/dt = x is uniquely defined by the initial
conditions (that is that the corresponding integral curves in the
(t,x)-plane do not intersect each other). This conclusion of the
mathematical model bears little relevance to the reality. A computer
experiment shows that all these integral curves have common points on
the negative t-semi-axis. Indeed, say, curves with the initial
conditions x(0) = 0 and x(0) = 1 practically intersect at t = -10
and at t = -100 you cannot fit in an atom between them. Properties of
the space at such small distances are not described at all by
Euclidean geometry. Application of the uniqueness theorem in this
situation obviously exceeds the accuracy of the model. This has to
be respected in practical application of the model, otherwise one
might find oneself faced with serious troubles.
I would like to note, however, that the same uniqueness theorem
explains why the closing stage of mooring of a ship to the quay
is carried out manually: on steering, if the velocity of approach
would have been defined as a smooth (linear) function of the distance,
the process of mooring would have required an infinitely long period
of time. An alternative is an impact with the quay (which is damped
by suitable non-ideally elastic bodies). By the way, this problem had
to be seriously confronted on landing the first descending apparata on
the Moon and Mars and also on docking with space stations - here the
uniqueness theorem is working against us.
Unfortunately, neither such examples, nor discussing the danger of
fetishising theorems are to be met in modern mathematical textbooks,
even in the better ones. I even got the impression that scholastic
mathematicians (who have little knowledge of physics) believe in the
principal difference of the axiomatic mathematics from modelling
which is common in natural science and which always requires the
subsequent control of deductions by an experiment.
Not even mentioning the relative character of initial axioms, one cannot
forget about the inevitability of logical mistakes in long arguments
(say, in the form of a computer breakdown caused by cosmic rays or
quantum oscillations). Every working mathematician knows that if one
does not control oneself (best of all by examples), then after some
ten pages half of all the signs in formulae will be wrong and twos
will find their way from denominators into numerators.
The technology of combatting such errors is the same external control by
experiments or observations as in any experimental science and it should
be taught from the very beginning to all juniors in schools.
Attempts to create "pure" deductive-axiomatic mathematics have led to
the rejection of the scheme used in physics (observation - model -
investigation of the model - conclusions - testing by observations) and
its substitution by the scheme: definition - theorem - proof. It is
impossible to understand an unmotivated definition but this does not
stop the criminal algebraists-axiomatisators. For example, they would
readily define the product of natural numbers by means of the long
multiplication rule. With this the commutativity of multiplication
becomes difficult to prove but it is still possible to deduce it as
a theorem from the axioms. It is then possible to force poor students
to learn this theorem and its proof (with the aim of raising the
standing of both the science and the persons teaching it). It is
obvious that such definitions and such proofs can only harm the
teaching and practical work.
It is only possible to understand the commutativity of multiplication by
counting and re-counting soldiers by ranks and files or by calculating
the area of a rectangle in the two ways. Any attempt to do without this
interference by physics and reality into mathematics is sectarianism and
isolationism which destroy the image of mathematics as a useful human
activity in the eyes of all sensible people.
I shall open a few more such secrets (in the interest of poor students).
The determinant of a matrix is an (oriented) volume of the
parallelepiped whose edges are its columns. If the students are told
this secret (which is carefully hidden in the purified algebraic
education), then the whole theory of determinants becomes a clear
chapter of the theory of poly-linear forms. If determinants are defined
otherwise, then any sensible person will forever hate all the
determinants, Jacobians and the implicit function theorem.
What is a group? Algebraists teach that this is supposedly a set with
two operations that satisfy a load of easily-forgettable axioms.
This definition provokes a natural protest: why would any sensible
person need such pairs of operations? "Oh, curse this maths" -
concludes the student (who, possibly, becomes the Minister for
Science in the future).
We get a totally different situation if we start off not with the group
but with the concept of a transformation (a one-to-one mapping of a set
onto itself) as it was historically. A collection of transformations of
a set is called a group if along with any two transformations it
contains the result of their consecutive application and an inverse
transformation along with every transformation.
This is all the definition there is. The so-called "axioms" are in fact
just (obvious) properties of groups of transformations. What axiomatisators
call "abstract groups" are just groups of transformations of various sets
considered up to isomorphisms (which are one-to-one mappings preserving the
operations). As Cayley proved, there are no "more abstract" groups in the
world. So why do the algebraists keep on tormenting students with the
abstract definition?
By the way, in the 1960s I taught group theory to Moscow
schoolchildren. Avoiding all the axiomatics and staying as close as
possible to physics, in half a year I got to the Abel theorem on the
unsolvability of a general equation of degree five in radicals
(having on the way taught the pupils complex numbers, Riemann surfaces,
fundamental groups and monodromy groups of algebraic functions).
This course was later published by one of the audience, V. Alekseev,
as the book The Abel theorem in problems.
What is a smooth manifold? In a recent American book I read that
Poincare was not acquainted with this (introduced by himself) notion
and that the "modern" definition was only given by Veblen in the late
1920s: a manifold is a topological space which satisfies a long
series of axioms.
For what sins must students try and find their way through all these
twists and turns? Actually, in Poincare's Analysis Situs there is
an absolutely clear definition of a smooth manifold which is much
more useful than the "abstract" one.
A smooth k-dimensional submanifold of the Euclidean space R^N is its
subset which in a neighbourhood of its every point is a graph of a
smooth mapping of R^k into R^(N - k) (where R^k and R^(N - k)
are coordinate subspaces). This is a straightforward generalization
of most common smooth curves on the plane (say, of the circle
x^2 + y^2 = 1) or curves and surfaces in the three-dimensional space.
Between smooth manifolds smooth mappings are naturally defined.
Diffeomorphisms are mappings which are smooth, together with their
inverses.
An "abstract" smooth manifold is a smooth submanifold of a Euclidean
space considered up to a diffeomorphism. There are no "more abstract"
finite-dimensional smooth manifolds in the world (Whitney's theorem).
Why do we keep on tormenting students with the abstract definition?
Would it not be better to prove them the theorem about the explicit
classification of closed two-dimensional manifolds (surfaces)?
It is this wonderful theorem (which states, for example, that any compact
connected oriented surface is a sphere with a number of handles) that
gives a correct impression of what modern mathematics is and not the
super-abstract generalizations of naive submanifolds of a Euclidean space
which in fact do not give anything new and are presented as achievements
by the axiomatisators.
The theorem of classification of surfaces is a top-class mathematical
achievement, comparable with the discovery of America or X-rays.
This is a genuine discovery of mathematical natural science and it is
even difficult to say whether the fact itself is more attributable to
physics or to mathematics. In its significance for both the applications
and the development of correct Weltanschauung it by far surpasses such
"achievements" of mathematics as the proof of Fermat's last theorem or
the proof of the fact that any sufficiently large whole number can be
represented as a sum of three prime numbers.
For the sake of publicity modern mathematicians sometimes present such
sporting achievements as the last word in their science. Understandably
this not only does not contribute to the society's appreciation of
mathematics but, on the contrary, causes a healthy distrust of the
necessity of wasting energy on (rock-climbing-type) exercises with
these exotic questions needed and wanted by no one.
The theorem of classification of surfaces should have been included in
high school mathematics courses (probably, without the proof) but for
some reason is not included even in university mathematics courses
(from which in France, by the way, all the geometry has been banished
over the last few decades).
The return of mathematical teaching at all levels from the scholastic
chatter to presenting the important domain of natural science is an
espessially hot problem for France. I was astonished that all the best
and most important in methodical approach mathematical books are
almost unknown to students here (and, seems to me, have not been
translated into French). Among these are Numbers and figures by
Rademacher and Toeplitz, Geometry and the imagination by Hilbert
and Cohn-Vossen, What is mathematics? by Courant and Robbins,
How to solve it and Mathematics and plausible reasoning by Polya,
Development of mathematics in the 19th century by F. Klein.
I remember well what a strong impression the calculus course by Hermite
(which does exist in a Russian translation!) made on me in my school years.
Riemann surfaces appeared in it, I think, in one of the first lectures
(all the analysis was, of course, complex, as it should be). Asymptotics
of integrals were investigated by means of path deformations on Riemann
surfaces under the motion of branching points (nowadays, we would have
called this the Picard-Lefschetz theory; Picard, by the way, was
Hermite's son-in-law - mathematical abilities are often transferred by
sons-in-law: the dynasty Hadamard - P. Levy - L. Schwarz - U. Frisch
is yet another famous example in the Paris Academy of Sciences).
The "obsolete" course by Hermite of one hundred years ago (probably, now
thrown away from student libraries of French universities) was much more
modern than those most boring calculus textbooks with which students are
nowadays tormented.
If mathematicians do not come to their senses, then the consumers who
preserved a need in a modern, in the best meaning of the word, mathematical
theory as well as the immunity (characteristic of any sensible person)
to the useless axiomatic chatter will in the end turn down the services
of the undereducated scholastics in both the schools and the universities.
A teacher of mathematics, who has not got to grips with at least some of
the volumes of the course by Landau and Lifshitz, will then become a
relict like the one nowadays who does not know the difference between an
open and a closed set.
V.I. Arnold
Translated by A.V. GORYUNOV